Lesson Study Plans by z4eM52K3

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									                                  Title: Pizza Topping Combinations

by Jerry Ruel, Zari Robinson, Greg Harris, Angela Stevens, Julia Valonis, Melissa Bell, and Beth Baldwin

                                  Pre-lesson date: November 13, 2008




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Research Aim:

Students will grow into persistent and flexible problem solvers.

Broad Content Goal:

Students will explore algebra concepts and communicate their mathematical ideas clearly and respectfully.

Lesson Objectives :
5.20 The student will analyze the structure of numerical and geometric patterns (how they change or grow) and express the relationship, using
      words, tables, graphs or a mathematical sentence. Concrete materials and calculators will be used.
5.21     The student will
         a) investigate and describe the concept of variable;
         b) use a variable expression to represent a given verbal quantitative expression involving one operation; and
         c) write an open sentence to represent a given mathematical relationship, using a variable.

Students will:

      represent data through physical and graphical means,
      draw conclusions from the data,
      communicate their findings to fellow classmates,
      explore the mathematical idea of combinations of two items,
      discuss whether or not order matters when determining combinations.

Lesson Overview:

Students will be given the following problem:

Tony’s Pizza is having a fundraiser to benefit Poplar Tree School. You can order pizzas with 3 different toppings: Pepperoni, Sausage, and
Mushroom. Your task is to determine the number of possible ways there are to order a pizza.

Repeat the problem using four toppings and then 5 toppings. Can you build a rule for determining the number of pizzas you can create with any
given number of toppings?


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Steps        Instructional activities                       Anticipated Student Responses                      Remarks on Teaching




Introduction Tell students we are going to be doing a       Students will be reminded of the process           Students must have done the
             problem involving combinations of items.       involved in solving a combination problem.         handshake problem previously
             Remind students about the handshake
             problem. Have 3 students model the             They might be wondering how this problem           Partners should be predetermined
             handshake problem. Record the number of        will relate to the handshake problem               before starting so they are ready to
             handshakes. Repeat with 4 students.                                                               collaborate when it is time



Engage       Ask the students to raise their hand if they      Students will be excited about doing a              Students should have
             like pizza.                                       problem about pizza.                                experience working in
                                                                                                                   cooperative groups.
             In 4 person groups, discuss their favorite        If interest is high, motivation, behavior and
             toppings for about 2 minutes.                     performance will be high.                           Need to review the rules for
                                                                                                                   respectful group work and
             Whole group: Brainstorm some pizza                Since it is connected to a previous problem         signals for wrapping up and
             toppings. Record on the SmartBoard.               which they completed, they should feel              freezing.
                                                               confident that they can succeed in solving
             Imagine that you have to order pizzas for         this problem.                                       Remind students there is more
             the whole class and you have to please                                                                than one way to solve a
             every person. How many pizzas do you              Some students who were confused by or               problem and that we are going
             think we would have to order?                     absent for the handshake problem may be             to explore the problem in
                                                               anxious about the new problem.                      groups and we will support
                                                                                                                   each other.




                                                                                                                                             3
Posing    Pass out the problem sheet. Have student            Creat pizza crust and toppings
Problem   read the problem silently. Students                 model on the BlackBoard
          highlight what they think will be key
          words or numbers as they read.                      Consider the reading level/English
          Then the teacher reads the problem out              language proficiency for reading
          loud from the Smartboard, highlighting              and interpreting the problems.
          key words that the students chose.


          Problem:

          Tony’s Pizza is having a fundraiser to benefit
          Poplar Tree School. You can order pizzas with 3
          different toppings: Pepperoni, Sausage, and
          Mushroom. Your task is to determine the number
          of possible ways there are to order a pizza.

          *Your count should include a choice of: no
          topping, and there cannot be double toppings (ex.
          double pepperoni).

          How many combinations can be made using the 3
          toppings?

          Suppose Tony decides to offer 4 toppings. How
          many combinations can be made?

          5 toppings?

          You may use paper and pencil or any available
          manipulatives to solve the problem. Show evidence
          of your thinking process on the recording sheet.
          (numbers, pictures, charts, words…)




                                                                                           4
                                                         Does order matter when placing the toppings
           Teacher uses pizza model on the               on the pizza? For example, is pepperoni-
           SmartBoard to show how to form the            mushroom the same as mushroom-
           different combinations using one and two      pepperoni? Allow time for students to
           toppings.                                     discuss this in their groups and come to a
                                                         consensus.
           Tell students they will be doing a similar
           problem, except with more toppings.           For the purposes of probability, order does
                                                         not matter when placing toppings on a pizza,
                                                         so these are the following possible
           Does anyone want to make a prediction for     combinations:
           how many choices of pizzas you could
           make with 3 toppings? 4 toppings?                 1.   No Topping
                                                             2.   Pepperoni
                                                             3.   Sausage                                 Students should have experience
                                                             4.   Pepperoni and Sausage                   managing math manipulatives

                                                         Some students may use multiplication to          Students will have previously
                                                         predict the results.                             learned some of the Multiple
                                                         2 topping = 4 pizzas (2 x 2 = 4) so              Representation models (Concrete,
                                                         3 topping = 6 pizzas (3 x 2)                     making tables, drawing pictures,
                                                                                                          verbalizing)
                                                         Some may square (using multiplication) the
                                                         number of toppings                               Pre cut and bag topping and crusts
                                                         2 topping = 4 pizzas (2 x 2 = 4) so
                                                         3 topping = 9 pizzas (3 x 3= 9)
Active     Discuss with your partner what kind of                                                         Teachers should check in with
                                                         (This is not correct, but don’t tell students)
Learning   manipulatives you would like to use. One                                                       each group as they get started to
           student will go to pick them up.             Possible Student Responses:                       see if they need help choosing or
                                                                                                          following through on a choice.
           Have Available:                              Students may want to change materials after
                                                        seeing what other groups are doing.               They should be allowed to change
               Crust circles                                                                             if they want to try another method.
               Bags of paper toppings—16 of
                each (4 toppings in separate bags)

                                                                                                                                       5
    Colored stickers
    Colored chips
    Unifix cubes
    Variety of manipulatives from
     math shelves
    Poster paper



Students Collaborate about materials they
want to use.

                                            Some students may build concrete models using Remind them to make one with no
                                            the crust circles and cut out toppings.       toppings and one with all toppings.

                                                  No                                       When using 3 toppings, it gets
                                                  topping                                  confusing to remember which
                                                                                           combinations they have already
                                                                                           used. Ask how they will record
                                                                                           their results so they can keep
                                                                                           track.




                                            Some will draw pictures using letters for
                                            toppings,


                                                  N          P           S          PS
                                                  o          o           N
                                                                         o
                                                                                     N o
                                                                         o           o




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Some will make a list or diagram

          None

          Pepperoni (or P)

          Sausage (or S)

          Pepperoni and Sausage ( or PS)

Some will tally as they create combinations.

1 Topping 2 Toppings 3 toppings

     ll           llll          llll lll



Encourage them to record connections between
the number of toppings and the number of
combinations.

Examples:

1 topping gives me 2 pizzas

2 toppings gives me 4 pizzas

3 toppings gives me 8 pizzas

 Toppings          Combinatio
                   ns
 1                 2
 2                 4
 3                 8

                                               7
                                                             Some students may create an advanced         If a pair/group finds a solution
                                                             table such as this.                          right away, encourage them to
                                                                                                          find another way to solve it.
                                                           #      #       0       1       2       3       Teachers go around the
                                                           top-   pizza   toppi   toppi   toppi   toppi   classroom and choose students
                                                           ping   s       ng      ng      ng      ng      work for the discussion that
                                                           s                                              follows (need good variety of
                                                             2      4       1       2       1             different combinations).
                                                             3      8       1       3       3        1
                                                             4     16       1       4       6        4
                                                             5     32       1       5      10       10
                                                             6     64       1       6      15       20
                                                             n     2n       1       n     [n(n-
                                                                                          1)]/2


Sharing     Sharing of student ideas: Students prepare a                                                  Are students attentive to
            way to share their solutions. Being guided                                                    others’ explanations?
            by teacher questions, students describe                                                       Use of different representations?
            their thinking as they found the solutions.
                                                                                                          Teacher lists different
                                                                                                          representations used.



Synthesis   Discuss:

            The multiple representations used

            Common findings—patterns?

            Algebraic thinking

            Rules built?


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             Connections to previous problems?
             (Handshake)
Evaluation   Through observations and                           Were students able to record the combinations in an
             discussions/sharing                                organized, understandable way?

                                                                Did students recognize a pattern? (Either doubling the
                                                                previous number of combinations or n2?)

                                                                Did students work well in their cooperative learning
                                                                groups?

                                                                Did students use multiple representations of their
                                                                algebraic thinking?

                                                                    Were students able to clearly share their methods
                                                                    with others?

Summary      With the list on the board with students’
             comments about relational patterns they
             noticed, students write in their journals (or on
             pieces of paper) what they have
             learned today, using any method they choose
             (words, numbers, pictures, etc.)




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